Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers
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Abstract (arXiv)
This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an ℓ1-penalty that involves some learnable dictionary; and (ii) an analysis form with an ℓ_∞-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted ℓ1 penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.
Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 29 of its 85 references that are in the map. First posted on arXiv 24 Mar 2025.
Builds on
29 of 85 references.
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17 more
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